Discrete Mathematics 2010-2022

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Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is bounding the maximum
\[
M(t):=\max_{0\leq H<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)\Big|
\]
of the absolute value of the incomplete sums $\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)$. In this very general context one of the most important results is the P\'olya-Vinogradov bound
\[
M(t)\leq \left\|\hat{t}\right\|_{\infty}\log 3p,
\]
where $\hat{t}:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of $t$. In this talk, we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that there exists a subset of $a\in\mathbb{F}_{p}^{\times}$ such that
\[
M( e((ax+\overline{x})/p))\geq \left(\frac{2}{\pi}+o(1)\right)\log\log p,
\]
as $p\rightarrow \infty$. We prove this by studying the growth of the moments of $\{M(e((ax+\overline{x})/p))\}_{a\in\mathbb{F}_{p}^{\times}}$. This is a joint work with Pascal Autissier and Youness Lamzouri.

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